Harmonic morphisms $\phi:(M,g)\to (N,h)$ are maps between Riemannian manifolds pulling back locally defined real-valued harmonic functions on $(N,h)$ to harmonic functions on $(M,g)$. They are solutions to an over-determined non-linear system of partial differential equations, heavily depending on the geometry of the manifolds involved. This means that they are difficult to find and do not even exist in simple cases.

In this lecture we give a brief introduction to the general theory and then focus on the special case when the codomain $(N,h)$ is the standard Euclidean complex plane $\mathbb C$. In that case a regular fibre $\phi^{-1}(\{z_0\})$ of a harmonic morphism $\phi:(M,g)\to\mathbb C$ is a minimal submanifold of the domain $(M,g)$ of codimension two. This is the primary reason for our interest in this mathematical problem.

We will introduce our "Method of Eigenfamilies" and show how this has proven the existence of solutions to this over-determined non-linear problem for all the Riemannian symmetric spaces.